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Properties

Label 1-1033-1033.1015-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.0230 + 0.999i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0851 + 0.996i)2-s + (−0.413 + 0.910i)3-s + (−0.985 − 0.169i)4-s + (0.728 − 0.685i)5-s + (−0.872 − 0.489i)6-s + (0.905 + 0.424i)7-s + (0.252 − 0.967i)8-s + (−0.658 − 0.752i)9-s + (0.620 + 0.783i)10-s + (0.760 + 0.648i)11-s + (0.561 − 0.827i)12-s + (0.658 + 0.752i)13-s + (−0.5 + 0.866i)14-s + (0.322 + 0.946i)15-s + (0.942 + 0.334i)16-s + (−0.847 + 0.531i)17-s + ⋯
L(s)  = 1  + (−0.0851 + 0.996i)2-s + (−0.413 + 0.910i)3-s + (−0.985 − 0.169i)4-s + (0.728 − 0.685i)5-s + (−0.872 − 0.489i)6-s + (0.905 + 0.424i)7-s + (0.252 − 0.967i)8-s + (−0.658 − 0.752i)9-s + (0.620 + 0.783i)10-s + (0.760 + 0.648i)11-s + (0.561 − 0.827i)12-s + (0.658 + 0.752i)13-s + (−0.5 + 0.866i)14-s + (0.322 + 0.946i)15-s + (0.942 + 0.334i)16-s + (−0.847 + 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0230 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0230 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.0230 + 0.999i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.0230 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078512265 + 1.103620690i\)
\(L(\frac12)\) \(\approx\) \(1.078512265 + 1.103620690i\)
\(L(1)\) \(\approx\) \(0.8899120969 + 0.6488572956i\)
\(L(1)\) \(\approx\) \(0.8899120969 + 0.6488572956i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.0851 + 0.996i)T \)
3 \( 1 + (-0.413 + 0.910i)T \)
5 \( 1 + (0.728 - 0.685i)T \)
7 \( 1 + (0.905 + 0.424i)T \)
11 \( 1 + (0.760 + 0.648i)T \)
13 \( 1 + (0.658 + 0.752i)T \)
17 \( 1 + (-0.847 + 0.531i)T \)
19 \( 1 + (0.601 - 0.798i)T \)
23 \( 1 + (-0.0121 - 0.999i)T \)
29 \( 1 + (0.950 - 0.311i)T \)
31 \( 1 + (-0.229 - 0.973i)T \)
37 \( 1 + (-0.181 - 0.983i)T \)
41 \( 1 + (0.981 + 0.193i)T \)
43 \( 1 + (0.776 - 0.630i)T \)
47 \( 1 + (-0.711 - 0.702i)T \)
53 \( 1 + (-0.957 - 0.288i)T \)
59 \( 1 + (0.950 + 0.311i)T \)
61 \( 1 + (0.744 + 0.667i)T \)
67 \( 1 + (-0.806 + 0.591i)T \)
71 \( 1 + (0.728 + 0.685i)T \)
73 \( 1 + (0.181 - 0.983i)T \)
79 \( 1 + (-0.942 + 0.334i)T \)
83 \( 1 + (0.711 - 0.702i)T \)
89 \( 1 + (0.791 + 0.611i)T \)
97 \( 1 + (-0.925 - 0.379i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.379867372363857273466221655269, −20.553976437287993993731835888962, −19.757166013505062402643399295541, −19.00405517096579012761829282512, −18.14697829926374926254301832319, −17.73259311078074951485675593749, −17.28031493928192963124836979804, −16.09124512470690746882440323022, −14.51663745073589867296157865151, −13.97429903724828680734345157486, −13.52786179935133802785298098012, −12.59258513726362712835498254218, −11.57627383364324571026159243819, −11.11533112524493137449308063123, −10.50191215893386079583662670702, −9.42146944358312182457568731894, −8.41885779467204382881679557256, −7.68680882366132223888094008385, −6.608187794338127497910068752291, −5.703726872780754312402321255830, −4.902201551583208927861049432201, −3.54676928258203363826578907766, −2.72350249152903502712925327523, −1.549844039724130534083032307304, −1.10777118784136735799635712616, 0.94672089234073202950459195782, 2.2114571062288955814981363098, 4.13985552851267262307996433138, 4.45475963938624124526950854205, 5.33514722359682493151626657527, 6.11248515254053934864187562634, 6.840525014365547215635391888251, 8.26915973273586764183105953094, 8.98963797798719015747242366034, 9.340239792908456394158962342213, 10.374949527522580511356263318672, 11.357040622934308657025193128381, 12.25847576930376053551251208181, 13.265390403875403872475181048942, 14.24286070616833243590428513637, 14.710527039450079641562153990324, 15.65689403045688993462456614897, 16.25603774627213830964335610878, 17.08025170222201173906779372090, 17.70503688764770734217902625635, 18.04502633676886780024536092591, 19.41517117367067806969563769814, 20.48012469791374093283301079593, 21.14102066728531263848999093890, 21.92460716864629265503729068296

Graph of the $Z$-function along the critical line